A Min–Max Property of Chordal Bipartite Graphs with Applications

We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E ( G ) equals the chromatic number of the complement of the square of line graph of G . Using this, we establish that for a chordal bipartite graph G...

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Bibliographic Details
Published inGraphs and combinatorics Vol. 26; no. 3; pp. 301 - 313
Main Authors Abueida, Atif, Busch, Arthur H., Sritharan, R.
Format Journal Article
LanguageEnglish
Published Japan Springer Japan 01.05.2010
Springer Nature B.V
Subjects
Algorithms
Combinatorics
Engineering Design
Graph theory
Graphs
Mathematics
Mathematics and Statistics
Original Paper
Polynomials
Chain cover
Bipartite graphs
Induced matching
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Summary:We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E ( G ) equals the chromatic number of the complement of the square of line graph of G . Using this, we establish that for a chordal bipartite graph G , the minimum number of chain subgraphs of G needed to cover E ( G ) equals the size of a largest induced matching in G , and also that a minimum chain subgraph cover can be computed in polynomial time. The problems of computing a minimum chain cover and a largest induced matching are NP-hard for general bipartite graphs. Finally, we show that our results can be used to efficiently compute a minimum chain subgraph cover when the input is an interval bigraph.
Bibliography:ObjectType-Article-2
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ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-010-0922-0